Question

What is the relationship between the acid ionization constant for a weak acid \(\left(K_{2}\right)\) and the base ionization constant for its conjugate base \(\left(K_{\mathrm{b}}\right) ?\)

Short answer

The relationship between the acid ionization constant \(K_a\) for a weak acid and the base ionization constant \(K_b\) for its conjugate base is given by the equation \(K_w = K_a \times K_b\), where \(K_w\) is the autoionization constant of water (1.0 x 10^-14 at 25°C).

Step-by-step solution

Understand the Relationship Between Ka and Kb

For a given weak acid and its conjugate base, the acid ionization constant (Ka) and the base ionization constant (Kb) are related by the autoionization constant of water (Kw). The relationship is given by the equation Kw = Ka * Kb. Since Kw is a constant at a given temperature (approximately 1.0 x 10^-14 at room temperature), if you know one of the constants, you can calculate the other by rearranging the equation.

Rearrange the Equation to Show Relationship

To find the relationship between Ka for a weak acid and Kb for its conjugate base, rearrange the equation to solve for Kb: Kb = Kw / Ka. This equation showcases that as the strength of the weak acid increases (higher Ka), the strength of its conjugate base decreases (lower Kb), and vice versa.

Conclusion

In conclusion, the product of the Ka for a weak acid and the Kb for its conjugate base equals the autoionization constant of water, Kw. This means that the dissociation constants for a conjugate acid-base pair are inversely related; when one is known, the other can be calculated.

Key concepts

Ka and Kb Relationship

Understanding the relationship between the acid ionization constant (\(K_a\)) and the base ionization constant (\(K_b\)) is crucial for students studying chemistry. To put it simply, for any given weak acid and its conjugate base, the values of \(K_a\) and \(K_b\) are interconnected by the autoionization constant of water (\(K_w\)). At room temperature, \(K_w\) is typically valued at approximately 1.0 x 10^-14.

If we look at the product of the \(K_a\) of a weak acid and the \(K_b\) of its conjugate base, it will always equal \(K_w\). So if the \(K_a\) of a weak acid is known, its conjugate base's \(K_b\) can easily be determined and vice versa. This is because \(K_b = K_w / K_a\). This relationship indicates an inverse proportionality; as the strength of a weak acid increases, the strength of its conjugate base decreases, suggesting that they are indeed two sides of the same coin.

Autoionization of Water

The autoionization of water concept is central to understanding the nature of acids and bases. Water, a neutral substance, can undergo a process known as autoionization where two water molecules react to form a hydronium ion (\(H_3O^+\)) and a hydroxide ion (\(OH^-\)). The equilibrium constant for this reaction is known as the autoionization constant, \(K_w\), and it plays a pivotal role in the acid-base chemistry.

At a given temperature, \(K_w\) remains constant, which means it provides a fundamental point of reference when evaluating the strengths of acids and bases. It's the product of the concentrations of the ions produced by water's autoionization: \[ K_w = [H_3O^+][OH^-]. \] The immutable nature of \(K_w\) under constant temperature conditions allows chemists to predict the behavior of acids and bases in aqueous solutions.

Conjugate Acid-Base Pair

A conjugate acid-base pair consists of two species that transform into each other by the gain or loss of a proton (\(H^+\)). In a chemical reaction, when a base receives a proton, it becomes its conjugate acid, and when an acid donates a proton, it becomes its conjugate base. This concept is essential for understanding not just the nature of substances, but also buffer solutions and the concept of equilibrium in acid-base reactions.

For example, when ammonia (\(NH_3\)) acts as a base and gains a proton, it forms ammonium (\(NH_4^+\)), its conjugate acid. Simultaneously, water, acting as an acid by donating a proton, becomes hydroxide (\(OH^-\)), its conjugate base. The conjugate pairs in this reaction are (\(NH_3/NH_4^+\)) and (\(H_2O/OH^-\)). Recognizing these pairs helps students predict the direction of acid-base reactions and calculate the pH of solutions.

Weak Acid and Conjugate Base

In the case of a weak acid, its dissociation in water is not complete, and only a small fraction of its molecules donate protons to form the conjugate base. A weak acid (\(HA\)) in water partially dissociates to produce hydrogen ions (\(H^+\)) and its conjugate base (\(A^-\)). Acid strength is quantified by the \(K_a\) value, which is the equilibrium constant for this dissociation reaction. A higher \(K_a\) value indicates a stronger weak acid, meaning it donates protons more readily.

The conjugate base formed from a weak acid tends to be a weak base, and its ability to accept protons is quantified by the \(K_b\) value. Understanding this relationship helps explain how changes in the concentration of various species within a solution can affect the overall pH. For example, adding more \(HA\) will increase the concentration of \(H^+\text{ and } A^-\text{ and } vice versa\). Knowing the relationship between a weak acid and its conjugate base is valuable for predicting the extent of chemical reactions and designing systems with desired acid-base properties.